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We present a new technique for the numerical simulation of axisymmetricsystems. This technique avoids the coordinate singularities which often arisewhen cylindrical or polar-spherical coordinate finite difference grids areused, particularly in simulating tensor partial differential equations likethose of 3+1 numerical relativity. For a system axisymmetric about the z axis,the basic idea is to use a 3-dimensional Cartesian (x,y,z) coordinate gridwhich covers (say) the y=0 plane, but is only onefinite-difference-molecule--width thick in the y direction. The field variablesin the central y=0 grid plane can be updated using normal (x,y,z)--coordinatefinite differencing, while those in the y \neq 0 grid planes can be computedfrom those in the central plane by using the axisymmetry assumption andinterpolation. We demonstrate the effectiveness of the approach on a set offully nonlinear test computations in 3+1 numerical general relativity,involving both black holes and collapsing gravitational waves.Comment: 17 pages, 4 figure

SYMMETRY WITHOUT SYMMETRY: NUMERICAL SIMULATION OF AXISYMMETRIC SYSTEMS USING CARTESIAN GRIDS